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%I #13 Apr 05 2019 17:45:19
%S 2,5,29,691,773,971,1217,1613,2207,2347,2791,3467,3491,6079,27073,
%T 45281,55609
%N Denominators of lower primes-only best approximates (POBAs) to sqrt(2); see Comments.
%C Suppose that x > 0. A fraction p/q of primes is a lower primes-only best approximate, and we write "p/q is in L(x)", if u/v < p/q < x < p'/q for all primes u and v such that v < q, where p' is least prime > p.
%C Let q(1) be the least prime q such that u/q < x for some prime u, and let p(1) be the greatest such u. The sequence L(x) follows inductively: for n > 1, let q(n) is the least prime q such that p(n)/q(n) < p/q < x for some prime p. Let q(n+1) = q and let p(n+1) be the greatest prime p such that p(n)/q(n) < p/q < x.
%C For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e The lower POBAs to sqrt(2) start with 2/2, 7/5, 41/29, 977/691, 1093/773, 1373/971. For example, if p and q are primes and q > 691, and p/q < sqrt(2), then 977/691 is closer to sqrt(2) than p/q is.
%t x = Sqrt[2]; z = 200; p[k_] := p[k] = Prime[k];
%t t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265776/A265777 *)
%t Numerator[tL] (* A265772 *)
%t Denominator[tL] (* A265773 *)
%t Numerator[tU] (* A265774 *)
%t Denominator[tU] (* A265775 *)
%t Numerator[y] (* A265776 *)
%t Denominator[y] (* A265777 *)
%Y Cf. A000040, A265759, A265772, A265774, A265775, A265776, A265777.
%K nonn,frac,more
%O 1,1
%A _Clark Kimberling_, Dec 20 2015
%E a(15)-a(17) from _Robert Price_, Apr 05 2019
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